Instrument-Based vs. Target-Based Rules∗
Marina Halac† Pierre Yared‡
April 15, 2020
Abstract
We study rules based on instruments vs. targets. Our application is a New Keynesian economy where the central bank has non-contractible information about aggregate demand shocks and cannot commit to policy. Incentives are provided to the central bank via punishment which is socially costly. Instrument- based rules condition incentives on the central bank’s observable choice of policy, whereas target-based rules condition incentives on the outcomes of policy, such as inflation, which depend on both the policy choice and realized shocks. We show that the optimal rule within each class takes a threshold form, imposing the worst punishment upon violation. Target-based rules dominate instrument- based rules if and only if the central bank’s information is sufficiently precise, and they are relatively more attractive the less severe the central bank’s commit- ment problem. The optimal unconstrained rule relaxes the instrument threshold whenever the target threshold is satisfied. Keywords: Policy Rules, Private Information, Delegation, Mechanism De- sign, Monetary Policy, Policy Objectives JEL Classification: D02, D82, E52, E58, E61
∗We would like to thank Hassan Afrouzi, V. V. Chari, John Cochrane, John Geanakoplos, Johannes H¨orner,Navin Kartik, Narayana Kocherlakota, Jen Jen La’O, Alessandro Lizzeri, George Mailath, Rick Mishkin, Ken Rogoff, Larry Samuelson, and Mike Woodford for comments. Jan Knoepfle and Angus Lewis provided excellent research assistance. †Yale University and CEPR: [email protected]. ‡Columbia University and NBER: [email protected]. 1 Introduction
The question of whether to base incentives on agents’ actions or the outcomes of these actions arises in various contexts. In particular, scholars and policymakers have long debated on the merits of using instrument-based vs. target-based rules to guide monetary policy.1 Instrument-based rules evaluate central bank performance on the choice of policy, taking a number of forms such as money growth rules, interest rate rules, and exchange rate rules. In contrast, target-based rules evaluate central bank performance on the outcomes of policy, such as the realized inflation level, price level, and output growth. Target-based rules have become increasingly popular over the last decades, with 60 central banks around the world adopting inflation targeting since 1990 (International Monetary Fund, 2019).2 Many of the early adopters were central banks in advanced economies, like those in New Zealand, Canada, and the United Kingdom, while more recent adopters included emerging economies such as India and Russia. Nevertheless, many developing economies still guide monetary policy based only on instruments; for example, many countries in Sub-Saharan Africa remain under an instrument-based rule in the form of a fixed exchange rate regime.3 As for the US, a number of studies describe US monetary policy as adhering to an instrument-based rule in the form of a Taylor rule from the 1980s through the early 2000s, while pursuing an inflation target since that time.4 In this paper, we develop a simple model to study and compare instrument-based and target-based rules. Our focus is on the implications of these rules for optimal incen- tives. While both rule classes are broadly used and discussed in the context of mone- tary policy, as just described, the theoretical literature on central bank incentives—and
1This debate includes many arguments for and against Taylor rules; see Svensson(2003), Bernanke(2004, 2015), McCallum and Nelson(2005), Appelbaum(2014), Blinder(2014), Taylor (2014), Kocherlakota(2016), and the Statement on Policy Rules Legislation signed by a num- ber of economists, available at http://web.stanford.edu/~johntayl/2016_pdfs/Statement_on_ Policy_Rules_Legislation_2-29-2016.pdf. There are also numerous discussions of inflation tar- geting; see Bernanke and Mishkin(1997), Bernanke et al.(1999), Mishkin(1999, 2017), Svensson (2003), and Woodford(2007), among others. 2This includes the 19 countries in the Eurozone plus 41 other countries classified by International Monetary Fund(2019) as inflation targeters. 3Of the 48 countries in sub-Saharan Africa, 21 have conventional pegs, 18 have managed exchange rates, and the rest have floating exchange rates. See International Monetary Fund(2019). 4See Kahn(2012), Taylor(2012), and Nikolsko-Rzhevskyy, Papell, and Prodan(2019) for the first time period and Bernanke(2012), Yellen(2012), and Shapiro and Wilson(2019) for the second one. The International Monetary Fund does not provide a classification of US monetary policy.
1 incentives in principal-agent relationships, more generally—has largely focused on one rule class or the other, lacking a general comparison that can inform this discussion. Using a mechanism design approach, we present a theory that elucidates the relative benefits of instrument-based vs. target-based rules and shows which class will be pre- ferred as a function of the environment. Additionally, we characterize the optimal unconstrained or hybrid rule and examine how combining instruments and targets can improve welfare. Our model builds on a canonical New Keynesian framework. Society delegates monetary policy to a central bank that lacks commitment and is therefore biased towards looser monetary policy and higher inflation at the time of choosing policy. The central bank’s policy choice is observable, but optimal policy depends on non- contractible information that the central bank possesses. Specifically, the central bank observes a private forecast of aggregate demand shocks to the economy (e.g. Romer and Romer, 2000), with a more negative forecast implying a looser socially optimal choice of monetary policy. Inflation and the output gap depend on both the central bank’s policy choice and the realized aggregate demand shock. Incentives are provided to the central bank via the threat of punishment. Punish- ment imposes a cost on both the central bank and society, stemming for example from formal sanctions, leadership replacement, or loss of central bank budget, autonomy, or reputation.5 We distinguish between different classes of rules depending on how incentives are structured: we say that the rule is instrument-based if punishments depend only on the central bank’s policy, and the rule is target-based if punishments depend only on the outcome of policy, namely realized inflation.6 Under either class of rule, punishments could be tailored so as to incentivize the central bank to choose the socially optimal policy, thus eliminating any distortions arising from the central bank’s inflation bias. However, because punishing the central bank is socially costly, society must trade off the benefit of mitigating its inflation bias with the benefit of
5See Rogoff(1985), Garfinkel and Oh(1993), and Walsh(1995, 2002) for a discussion of the differ- ent forms these punishments can take. In New Zealand the Governor of the Reserve Bank is subject to dismissal if he fails to achieve the inflation target (H¨ufner, 2004), and in several countries central- bank officials are subject to scrutiny of their policies with requirements of written explanations, or public hearings in the Parliament, or reviews by independent experts (Svensson, 2010). Consequences such as dismissal imply a cost not only to the central banker but to society at large. 6Our setting is one where the “divine coincidence” holds (e.g. Blanchard and Gal´ı, 2007), implying a one-to-one mapping between inflation and the output gap (holding fixed expectations of the future). Therefore, a target rule based on inflation is equivalent to one based on the output gap, and giving the central bank full discretion would be optimal absent an inflation bias.
2 reducing equilibrium punishments.7 Our analysis begins by showing that, within each class, an optimal rule takes a threshold form, with violation of the threshold leading to the worst feasible punish- ment. In the case of an optimal instrument-based rule, the central bank is allowed to choose any policy up to a threshold and is maximally punished for choosing looser policy. The idea is analogous to that in other models of delegation (reviewed below); since the central bank is inflation-biased relative to society, this punishment structure is optimal to deter the central bank from pursuing excessively expansionary policies. In the case of an optimal target-based rule, the central bank is subject to an inflation threshold, being maximally punished if realized inflation is above it. This punishment structure also incentivizes the central bank to not choose excessively expansionary policy, since this results in higher inflation in expectation. High-powered incentives of this form arise in moral hazard settings with hidden action.8 Our main result uses this characterization of the optimal rules for each class to compare their performance. We show that target-based rules dominate instrument- based rules if and only if the central bank’s private information is sufficiently precise. To illustrate, suppose the central bank has a perfect forecast of aggregate demand shocks. Then society guarantees the socially optimal policy by providing steep in- centives under a target-based rule, where punishments do not occur on path because the perfectly informed central bank chooses the policy that delivers the inflation tar- get outcome. This target-based rule strictly dominates any instrument-based rule, as the latter cannot incentivize the central bank while giving it enough flexibility to respond to its information. At the other extreme, suppose the central bank has no private information. Then society guarantees the ex-ante socially optimal policy with an instrument-based rule that ties the hands of the central bank, namely that punishes the central bank if any looser policy is chosen. This instrument-based rule strictly dominates any target-based rule, as the latter gives the central bank unnec- essary discretion and requires on-path punishments to provide incentives. Our main result proves that this logic applies more generally as we locally vary the precision of the central bank’s private information, away from the extremes of perfect and no information. 7Formally, punishment corresponds to “money burning” in principal-agent models of delegation (e.g., Amador and Bagwell, 2013). The agent in our setting is the central bank, and the principal is society, or equivalently the central bank’s ex-ante “self” which is not inflation-biased. 8See for example Innes(1990) and Levin(2003). Here these incentives arise because punishments cannot depend directly on the central bank’s policy under a target-based rule.
3 We additionally show that the benefit of using a target-based rule over an instrument- based rule is decreasing in the severity of the central bank’s commitment problem, namely decreasing in its inflation bias and increasing in the severity of punishment. Intuitively, the less biased is the central bank, the less costly is incentive provision under a target-based rule, as the central bank can be deterred from choosing exces- sively expansionary policy with less frequent punishments. Similarly, the harsher is the punishment imposed on the central bank for exceeding the inflation threshold, the less often are punishments required on path to implement a target outcome. We show that these two forces therefore make target-based rules more appealing than instrument-based rules on the margin. Our results shed light on the adoption of inflation targeting around the world. In particular, our analysis suggests that inflation targeting may be more suitable to advanced economies where the central bank commitment problem tends to be less severe compared to emerging market economies.9 Fraga, Goldfajn, and Minella(2003) indeed find that inflation targeting has been comparatively less successful in emerging markets, where deviations from the target have been larger and more common than in advanced economies. At the same time, our findings can also justify the more recent adoption of inflation targeting rules in emerging markets which have experienced an improvement in central bank independence. Viewing the central bank’s bias as fueled by political interests, such independence implies a smaller bias and thus greater benefits from adopting inflation targeting. A natural question from our analysis is how instruments and targets can be com- bined to improve upon the above rules that rely exclusively on one of these tools. We study the optimal hybrid rule, that is the optimal unconstrained rule in which punishments can depend freely on the central bank’s policy and realized inflation.10 Perhaps surprisingly, we show that this rule admits a simple form: an instrument threshold that is relaxed whenever a target threshold is satisfied. The optimal hybrid rule dominates instrument-based rules by allowing the central bank more flexibility to use expansionary policy under a target-based criterion, and it dominates target-based rules by more efficiently limiting the central bank’s discretion with direct punishments. An implementation of the optimal hybrid rule would be a Taylor rule which, whenever violated, switches to an inflation target. Notably, some policymakers and economists
9Relatedly, Mishkin(2000) and Taylor(2014) also conclude that inflation targeting may not be appropriate for many emerging-market countries. 10This rule yields the highest social welfare that can be achieved given the central bank’s inflation bias and private information.
4 advocated such an approach in the US in the aftermath of the Global Financial Crisis; see for example Yellen(2015, 2017) and the discussion in Walsh(2015). 11 This paper is related to several literatures. It relates to the literature on central bank credibility and reputation pioneered by Rogoff(1985), including work that ex- amines the role of private information in such a context.12 We follow Athey, Atkeson, and Kehoe(2005) by taking a mechanism design approach to characterize optimal policy subject to private information constraints. The paper also fits into the mecha- nism design literature that studies the tradeoff between commitment and flexibility in principal-agent delegation settings, building on the seminal work of Holmstr¨om(1977, 1984). In particular, in addition to Athey, Atkeson, and Kehoe(2005), the delegation models of Waki, Dennis, and Fujiwara(2018) on monetary policy and Amador, Wern- ing, and Angeletos(2006) and Halac and Yared(2014, 2018, 2019) on fiscal policy are closely related.13 Our main departure from these two literatures is that we distin- guish between incentives that are based on an agent’s actions (as in these literatures) and incentives that are based on observable noisy outcomes, and we characterize opti- mal incentives which can condition on both variables.14 Additionally, we depart from these literatures in that we allow for limited enforcement of rules, since incentives are provided via a finite punishment in our model.15 Prior work has considered the optimal choice of policy instrument, as well as the optimal choice of policy target. For a starting point on instruments, see Friedman (1960) on money growth rules, McCallum(1981) and Taylor(1993) on interest rate rules, and Devereux and Engel(2003) on exchange rate rules. Poole(1970) and Weitz- man(1974) provide a broad discussion of instrument rules based on prices—such as interest rates or exchange rates—versus instrument rules based on quantities—such
11The policy of the Italian government after it joined the European Exchange Rate Mechanism (ERM) in 1979 also resembled an optimal hybrid rule of this form. In 1992, the Italian government left the ERM to respond to its weak economy and devalue its currency. It subsequently pursued a more expansionary monetary policy that targeted inflation, until it returned to the ERM in 1996. See Bank of Italy(1996), Passacantando(1996), and Buttiglione, Giovane, and Gaiotti(1997). 12For example, see Barro and Gordon(1983a,b), Backus and Driffill(1985), Canzoneri(1985), Cukierman and Meltzer(1986), Walsh(1995), Sleet(2001), Albanesi, Chari, and Christiano(2003) and Kocherlakota(2016), among others. 13More generally, see Alonso and Matouschek(2008), Amador and Bagwell(2013), and Ambrus and Egorov(2017). 14Waki, Dennis, and Fujiwara(2018) provide an inflation-targeting implementation of their optimal interest-rate rule. 15This is important to assess target-based rules, which would otherwise achieve the socially optimal outcome with infinite punishments that occur with virtually zero on-path probability (cf. Mirrlees, 1974). Halac and Yared(2019) studies delegation under limited enforcement in a fiscal policy setting.
5 as money growth. Atkeson, Chari, and Kehoe(2007) show that the optimal choice of monetary policy instrument depends on tightness and transparency. For a study of policy targets, see the general framework of Giannoni and Woodford(2017), as well as an extensive body of work that examines inflation targeting, as noted in fn. 1. In con- trast to these literatures which consider either instrument-based or target-based rules, the goal of our paper is to compare the performance of these different rule classes, with a focus on their implications for incentive provision. A comparison of policy instruments and policy targets appears in the recent work of Angeletos and Sastry(2019), which examines the optimal form of forward guidance. Their focus however is on bounded rationality, namely households making mistakes in reasoning about others’ behavior and the equilibrium effects of policy. Instead, our focus is on the policymaker’s lack of commitment and how to optimally provide incentives given the resulting bias in preferences. By studying incentives under both private information and noisy outcomes, our paper also relates to the work of Riordan and Sappington(1988) and the literature that followed it. In the classic context of a regulator and a privately informed firm, the main question of this literature is whether the first best can be achieved by making transfers dependent on both the firm’s behavior and ex-post public signals correlated with the firm’s type.16 These models differ from ours in several aspects, most importantly in that incentives are provided via transfers, which are ruled out in our delegation environment.17 Moreover, while the focus of this literature is on the conditions under which the first best obtains under hybrid schemes, we characterize and compare second- best rules when costly punishments may condition only on behavior, or only on realized outcomes, or on both. Finally, our paper is related to other work that considers the use of socially costly punishments as incentives like we do, albeit in different environments. See for example Acemoglu and Wolitzky(2011) and Padr´oi Miquel and Yared(2012). 18
16The results of this literature are related to those of Cremer and McLean(1985) in auction theory. 17Some of these papers consider limited liability constraints; see Kessler, L¨ulfesmann,and Schmitz (2005) and references therein. Other differences with our work include that values are private in these models and, except for Gary-Bobo and Spiegel(2006), public signals are only informational. 18Although the settings and analyses differ in many aspects, our paper also relates to some models of career concerns for experts which study how the information a principal has affects reputational incentives. In particular, Prat(2005) distinguishes between information on actions and on outcomes.
6 2 Model
We present a monetary policy model in which the central bank observes a demand shock forecast and then chooses policy subject to lack of commitment. The forecast is the central bank’s private information, so we refer to it as the central bank’s type. Inflation and output depend on both the choice of monetary policy and the realized demand shock. Society provides incentives to the central bank via punishment which is socially costly. After describing the environment and our solution concept, we show that the model takes the form of a principal-agent delegation problem, with society being the principal and the central bank the agent.
Setup. Consider a linearized closed economy New Keynesian model with demand shocks (e.g. Benigno and Woodford, 2005). The economy consists of society and a central bank, and the horizon is infinite with time periods t = 0, 1,... The New Keynesian Phillips curve in period t, linking output to inflation, can be written as
πt = βEtπt+1 + κxt, (1) where πt ∈ R is the inflation rate, xt ∈ R is the output gap, β ∈ (0, 1) is the social discount factor, and κ > 0 is the slope of the Phillips curve. The Euler equation, linking output growth rates to real interest rates, is
xt = Etxt+1 − ζ(it − Etπt+1) − θt/κ, (2) where it ∈ R is the nominal interest rate, ζ > 0 is the elasticity of intertemporal substitution, and θt ∈ R is a zero-mean aggregate demand shock which we normalize by κ to simplify the notation. We assume that θt is independent and identically distributed (i.i.d.) over time; we show in Subsection 6.2 that our results extend to a setting in which θt is persistent. L H In every period t, the central bank privately observes a signal st ∈ {s , s } and 19 chooses the interest rate it. The demand shock θt is then realized, market expecta- tions of the future are formed, and inflation πt and the output gap xt are determined. We remark that while the policy instrument in our closed economy is the interest rate it, our results apply directly to an open economy in which the policy instrument is
19 Taking the signal st to be binary allows us to present our main insights in a transparent way. In Section 5, we extend our results to the case in which st is drawn from a continuum.
7 the exchange rate. As discussed in Subsection 6.3, the reason is that the uncovered interest rate parity condition implies a one-to-one mapping between the central bank’s choice of exchange rate and its choice of interest rate (Gal´ı, 2015). L H The central bank’s signal st ∈ {s , s } is informative about the demand shock. Specifically, we assume that the conditional distribution of the shock is normal with 2 mean equal to the signal, i.e., θt|st ∼ N (st, σ ). The precision of the central bank’s information is given by σ−1 > 0. We take sL = −∆ and sH = ∆ for some ∆ > 0 and assume that each signal occurs with equal probability. The shock’s unconditional distribution is thus a mixture of two normal distributions and has mean and variance given by 2 2 E(θ) = 0 and V ar(θ) = σ + ∆ .
Society observes the central bank’s policy it and the realized shock θt (or, equiv- alently, the central bank’s policy it and the realized inflation πt). Society cannot however deduce the central bank’s type st from these observations, as the distribution of θt has full support over the entire real line for each signal st. Society can incentivize the central bank by imposing penalties which are mutually costly to the central bank and society itself.20 We model these penalties by letting society commit ex ante to punishments Pt(·), specifying a cost to be imposed at the end of period t as a function of the central bank’s policy and/or realized inflation. As described below, our analysis will focus on Markov equilibria, so we require the punishment function Pt(·) to condition only on period-t variables, namely policy it 21 and/or inflation πt. Additionally, we require Pt ∈ [0, P ] for some finite P . As discussed in the Introduction, the punishments specified by society may represent formal penalties due to sanction regimes as well as informal penalties in the form of a loss of reputation for the central bank. Note that our assumption that the signal st is privately observed by the central bank can be equivalently interpreted as a restriction on the punishment function Pt(·), which cannot explicitly condition on the signal.
20Our model abstracts from any transfers between society and the central bank, which are rarely observed in practice and are also ruled out in the literature on principal-agent delegation. The punishment in our model being common to society and the central bank captures, in a stark way, the fact that providing the central bank with incentives is socially costly. In Section 6, we discuss an extension in which punishments harm society and the central bank asymmetrically. 21Observe that since punishments are socially costly, society’s rule will not be renegotiation proof, and thus commitment to Pt(·) is important. This commitment is ensured in practice via procedural rules designed to follow through on sanctions. Our analysis and results could be extended to allow for a small probability of renegotiation, with its effect being analogous to a reduction in P .
8 Payoffs and First Best. Expected welfare at date t is given by
∞ 2 2 X (κxt+k) π βk ακx − γ − (1 − γ) t+k − P , Et−1 t+k 2 2 t+k k=0 for some α > 0, γ ∈ [0, 1]. Substituting with the Phillips curve in (1), this can be rewritten in terms of the sequence of inflation:22
( ∞ " 2 2 # ) X (πt+k − β t+kπt+k+1) π απ + βk −γ E − (1 − γ) t+k − P . (3) Et−1 t 2 2 t+k k=0
Note that these preferences are time-inconsistent: for k ≥ 1, πt+k enters differently into welfare from the perspective of date t relative to the perspective of date t + k. We follow Benigno and Woodford(2005) by assuming that society cares about welfare from the “timeless perspective.” That is, social welfare is given by
∞ ( " 2 2 # ) X (πt − β tπt+1) π βt −γ E − (1 − γ) t − P . (4) E−1 2 2 t t=0
Instead, the central bank at date t wishes to maximize the welfare function in (3), reflecting the fact that the central bank lacks commitment to policy. To illustrate the commitment problem, consider the first-best policy that maximizes social welfare in (4). This policy prescribes no punishment, i.e. Pt = 0 for all t = 0, 1,..., and it implies a “divine coincidence” (e.g. Blanchard and Gal´ı, 2007) as social welfare is maximized with inflation stabilization. Plainly, under full commitment and perfect forecasting of the demand shocks θt, setting an interest rate it = −θt/(κζ) in each period t effectively counteracts the demand shocks, targeting zero inflation and a zero output gap at all dates. In fact, analogous logic applies if the central bank observes only an imperfect signal st of the shock θt but can commit to its policy choice.
In this case, the central bank would set it = −st/(κζ) in each period t in order to guarantee zero expected inflation and a zero expected output gap at all dates. Things however change when the central bank lacks commitment to policy. In this case, the central bank at date t maximizes its welfare represented by equation (3), which puts additional weight α > 0 on current inflation πt compared to the social welfare function in (4). The reason is that, lacking commitment, the central bank at
22 T This representation is derived under the condition that limT →∞ Et−1β πT = 0. As we will describe, we focus on Markov perfect equilibria where this condition holds.
9 t does not internalize the impact of current inflation on past inflation expectations. As a result, the central bank’s policy at t may differ from the socially optimal policy it = −st/(κζ). We will refer to α as the central bank’s inflation bias.
Equilibrium and Reformulation. Consider policy at date t under lack of com- mitment by the central bank, given a punishment specified by society as a function of the central bank’s policy choice it and/or realized inflation πt. In principle, the cen- tral bank’s equilibrium strategy could depend on the entire history of signals, demand shocks, inflation, output gaps, interest rates, and punishments, potentially leading to a complicated path of private sector expectations. To make our analysis tractable, we focus on Markov perfect equilibria in which the central bank’s strategy at date t depends only on its current signal st and the punishment function Pt(·), and in which private sector expectations of the future are constant, as is natural by the signals and shocks being i.i.d. over time.23 Under this solution concept, equilibrium policies and payoffs are stationary, and incentives are provided to the central bank via within-period punishments only.24 To simplify the notation, we thus remove time subscripts hereafter, and we reformulate the problem as follows. Observe that given constant expectations of future inflation Eπ and output gap Ex (both on- and off-the-equilibrium path), a central bank’s observable choice of interest rate i is equivalent to an observable choice of policy µ defined as
µ ≡ (β + κζ)Eπ + κEx − κζi.
Moreover, substituting into (1) and (2), inflation is then given by π = µ − θ, expected inflation by Eπ = Eµ, and the output gap by x = (µ − θ − βEµ)/κ. Using this notation, we can then redefine the punishment specified by society as a function of µ and π = µ − θ, namely a function P (µ, µ − θ). 2 2 Let φ(z|z, σz) be the normal density of a variable z with mean z and variance σz, 2 and let Φ(z|z, σz) be the corresponding normal cumulative distribution function. With the notation just introduced, the per-period social welfare corresponding to (4) can be
23We therefore ignore issues relating to potentially unstable expectations and implementation (see Woodford, 2003; Cochrane, 2011; Gal´ı, 2015). 24That is, we rule out incentives via self-enforcing continuation play. Halac and Yared(2020) studies optimal equilibria in a monetary model under history-dependent strategies, showing that incentive provision can require the economy to transition between low inflation and high inflation regimes over time.
10 written as
X 1 Z ∞ U(µj, θ, µ) − P (µj, µj − θ) φ(θ|sj, σ2)dθ, (5) 2 E j=L,H −∞ where
(µj − θ − β µ) 2 (µj − θ)2 U(µj, θ, µ) = −γ E − (1 − γ) . E 2 2
Recall that the central bank’s welfare differs from society’s due to the time incon- sistency problem. After observing its type sj, and taking as given the punishment function P (µj, µj − θ) and expectations Eµ, the central bank chooses a policy µj to maximize
Z ∞ j j j j j 2 α µ − θ + U(µ , θ, Eµ) − P (µ , µ − θ) φ(θ|s , σ )dθ (6) −∞ for j = L, H. With this reformulation, it is easy to view our setting as one of principal-agent delegation. Society is the principal and the central bank is the agent. Society has a preferred, or first-best, policy which is given by µfb(s) = s, implying zero expected inflation and a zero expected output gap, i.e., Eµ = Eπ = 0. The central bank is better informed than society and has a preferred, or flexible, policy which is given by µf (s, Eµ) = s + α + γβEµ.25 Thus, if granted full flexibility, the central bank’s policy would imply strictly positive expected inflation and a strictly positive expected output gap, i.e., Eµ = Eπ > 0. It is in this sense that the central bank is inflation-biased. We make two assumptions on parameters. First, we assume that ∆ is relatively small, so that the central bank types are relatively “close” to each other:
Assumption 1. α ≥ 2∆.
We make this assumption to ensure that our results do not rely on a discrete distance between the central bank types, as we verify in our extension to a continuum of types in Section 5. The implication of Assumption 1 is that, holding Eµ = 0, the central bank’s flexible policy exceeds the first-best policy under each signal:
sL < sH ≤ sL + α < sH + α. (7)
25Observe that if γ = 0, then our setting corresponds to a delegation setting with quadratic preferences and a constant bias (e.g., Melumad and Shibano, 1991; Alonso and Matouschek, 2008).
11 Second, we assume that society has a sufficient breadth of incentives to use in disciplining the central bank:
1 α 2 Assumption 2. P ≥ . 2φ(1|0, 1) 1 − γβ
Classes of Rules. We distinguish between different classes of rules according to how incentives are structured. We say that a rule is instrument-based if society commits to a punishment which depends only on the central bank’s policy choice µ, that is, a punishment function P (µ). A rule instead is target-based if society commits to a punishment that depends only on the realized inflation π = µ−θ, that is, a punishment function P (µ − θ). Finally, if society commits to a punishment function P (µ, µ − θ) that depends freely on µ and θ (and therefore freely on µ and π), we say that the rule is hybrid. We are interested in comparing the performance of these different classes of rules as the environment changes. Our analysis will consider varying the precision of the central bank’s private information while holding fixed the mean and variance of the shock θ. At one extreme, we can take σ → pV ar(θ) and ∆ → 0, so the central bank is uninformed with signal sL = sH = 0. At the other extreme, we can take σ → 0 and ∆ → pV ar(θ), so the central bank is perfectly informed with signal sj = θ.26 Note that since Assumption 1 holds for all feasible σ > 0 and ∆ > 0 given V ar(θ) fixed, the assumption implies α ≥ 2pV ar(θ) ≥ 2σ.
3 Instrument-Based and Target-Based Rules
We examine rules based on instruments versus targets. Subsection 3.1 and Subsec- tion 3.2 solve for the optimal rule within each class. Subsection 3.3 offers a comparison and shows that the socially optimal class of rule depends on the precision of the cen- tral bank’s private information, the central bank’s inflation bias, and the severity of punishment.
26Note that in our setting, welfare under an instrument-based or a target-based rule depends only on the mean and variance of θ. This avoids additional complications stemming from the fact that higher moments of the distribution of θ vary with σ and ∆.
12 3.1 Optimal Instrument-Based Rule
An instrument-based rule specifies a policy µj for each central bank type j = L, H and a punishment P (µ) as a function of the policy choice µ. Let P j ≡ P (µj) for j = L, H. The allocation {µL, µH ,P L,P H } must satisfy private information, enforcement, feasibility, and rationality constraints, as we describe next. The private information constraint captures the fact that the central bank can misrepresent its type. The rule must be such that, for j = L, H, a central bank of type j has no incentive to deviate privately from policy µj to policy µ−j:
Z ∞ j j j j 2 αµ + U(µ , θ, Eµ) − P φ(θ|s , σ )dθ −∞ Z ∞ −j −j −j j 2 ≥ αµ + U(µ , θ, Eµ) − P φ(θ|s , σ )dθ. (8) −∞
The enforcement constraint captures the fact that the central bank can freely choose any policy µe ∈ R, including policies not assigned to either type. The rule must be such that, for j = L, H, a central bank of type j has no incentive to deviate publicly j L H from policy µ to any policy µe ∈ / {µ , µ }: Z ∞ j j j j 2 αµ + U(µ , θ, Eµ) − P φ(θ|s , σ )dθ −∞ Z ∞ j 2 ≥ [αµe + U(µ,e θ, Eµ) − P (µe)] φ(θ|s , σ )dθ. −∞
Note that since the punishment satisfies P (µ) ≤ P for all µ ∈ R, the above inequality must hold under maximal punishment, i.e. when P (µ) = P . Moreover, since the inequality must then hold for all µe ∈ R, it must necessarily hold when µe corresponds to type j’s flexible policy µf (sj, Eµ) = sj + α + γβEµ. A necessary condition for the enforcement constraint to be satisfied is thus
Z ∞ j j j j 2 αµ + U(µ , θ, Eµ) − P φ(θ|s , σ )dθ −∞ Z ∞ f j f j j 2 ≥ αµ (s , Eµ) + U(µ (s , Eµ), θ, Eµ) − P φ(θ|s , σ )dθ (9) −∞ for j = L, H, where note that the right-hand side is the central bank’s minmax payoff. Constraints (8) and (9) are clearly necessary for an instrument-based rule prescrib- ing {µL, µH ,P L,P H } to be incentive compatible. Furthermore, if these constraints are
13 satisfied, then this allocation can be supported by specifying the worst punishment P following any policy choice µ∈ / {µL, µH }. Since such a choice is off path, it is without loss to assume that it is maximally punished. Finally, feasibility of the rule requires that for j = L, H,
P j ∈ [0, P ], (10) and rationality of private sector expectations requires
1 1 µ = µL + µH . (11) E 2 2
An optimal instrument-based rule maximizes expected social welfare subject to the private information, enforcement, feasibility, and rationality constraints:
Z ∞ X 1 j j j 2 max U(µ , θ, Eµ) − P φ(θ|s , σ )dθ (12) µL,µH ,P L,P H 2 j=L,H −∞
subject to, for j = L, H,(8), (9), (10), and (11).
Define a maximally-enforced instrument threshold µ∗ as a rule that prescribes no punishment if the central bank’s policy is weakly below a threshold µ∗ and the maximal punishment P if the policy exceeds this threshold. Such a rule can take the form of a maximally-enforced lower bound on the nominal interest rate in our environment.27 We find:
Proposition 1. The optimal instrument-based rule specifies µL = µH = 0 and P L = P H = 0. This rule can be implemented with a maximally-enforced instrument threshold µ∗ = 0.
The optimal instrument-based rule assigns both central bank types the policy that maximizes ex-ante social welfare. The central bank is given no discretion, and punish- ments occur only off path, if the central bank were to publicly deviate to a different policy. The simple structure of this rule will allow us to compare its performance to that of the optimal target-based rule in a transparent way. Our results however do not rely on this simple structure, as we show in Section 5.28
27In Subsection 6.2, we show that such a lower bound will be a function of the state in a setting in which demand shocks are persistent over time. 28In the setting of Section 5 with a continuum of signals, the optimal instrument-based rule does
14 It is worth noting that an instrument-based rule that induces the central bank to choose the first-best policy, µfb(sj) = sj for j = L, H, is available to society. Specifically, the low type can be dissuaded from choosing sH by specifying an on-path punishment P H > 0 (and both types can be dissuaded from choosing any policy µ∈ / {sL, sH } by specifying the worst punishment off path). However, because punishment is socially costly, Proposition 1 shows that such a rule is strictly dominated by a maximally-enforced instrument threshold µ∗ = 0. To prove Proposition 1, we solve a relaxed version of the program in (12) which ignores the private information constraint (8) for the high type and the enforcement constraints (9) for both types. We show that under Assumption 1, the solution to this relaxed problem entails no discretion, and it thus satisfies (8) for both types. Moreover, Assumption 2 guarantees that (9) is also satisfied for both types. Proposition 1 relates to the findings of an extensive literature on delegation, which provides conditions under which threshold delegation with no money burning is opti- mal. A general treatment can be found in Amador and Bagwell(2013). The analysis in Halac and Yared(2019) is also related in that it incorporates enforcement con- straints like those in (9) into a delegation framework, and it shows the optimality of maximally-enforced thresholds where on- and off-path violations lead to the worst punishment. In the current setting with binary signals, enforcement constraints are non-binding by Assumption 2, so punishments occur only off path. We address the issues that arise when enforcement constraints bind and punishments occur on path in our extension to a continuum of signals in Section 5.
3.2 Optimal Target-Based Rule
A target-based rule specifies a policy µj for each central bank type j = L, H and a punishment P (π) as a function of inflation π = µ − θ. Note that P (π) is defined L H for π ∈ R since θ is normally distributed. The allocation {µ , µ , {P (π)}π∈R} must satisfy incentive compatibility, feasibility, and rationality constraints, as we describe next. Incentive compatibility requires that the policy prescribed for each central bank type solve this type’s welfare-maximization problem. Given its private information and the punishment function specified by society, the central bank takes into account how provide discretion to the central bank and does involve punishments on path, yet our main insights continue to apply.
15 its policy affects the distribution of inflation and, thus, punishments. For j = L, H, µj must satisfy:
Z ∞ j j 2 µ ∈ arg max [αµe + U(µ,e θ, Eµ) − P (µe − θ)] φ θ|s , σ dθ . (13) µe −∞
Additionally, feasibility of the rule requires
P (π) ∈ 0, P for all π ∈ R, (14) and rationality of private sector expectations requires
1 1 µ = µL + µH . (15) E 2 2
An optimal target-based rule maximizes expected social welfare subject to the incentive compatibility, feasibility, and rationality constraints:
Z ∞ X 1 j j j 2 max U(µ , θ, Eµ) − P (µ − θ) φ θ|s , σ dθ (16) µL,µH ,{P (π)} 2 π∈R j=L,H −∞
subject to, for j = L, H,(13) and (14), and (15).
Note that integration by substitution yields
Z ∞ Z ∞ P (µ − θ)φ θ|sj, σ2 dθ = P (π)φ µ − sj − π|0, σ2 dπ, (17) −∞ −∞ where we have used the fact that φ (θ|s, σ2) = φ (θ − s|0, σ2) since φ (·) is the density of a normal distribution. Using (17) to substitute in (13), the first-order condition of the central bank’s problem is
Z ∞ j j 0 j j 2 α − (µ − s ) + γβEµ + P (π)φ µ − s − π|0, σ dπ = 0 for j = L, H. (18) −∞
Condition (18) is necessary for the rule to be incentive compatible. Its solution j j is µ = s + δ for j = L, H and some δ R 0, where δ is independent of the central bank’s type j. The latter observation allows us to simplify the problem in (16) as social welfare then also becomes independent of the central bank’s type j. Define a maximally-enforced target threshold π∗ as a rule that prescribes no pun- ishment if inflation is weakly below a threshold π∗ and the maximal punishment P if
16 inflation exceeds this threshold. We find:
Proposition 2. The optimal target-based rule specifies µj = sj +δ, P (π) = 0 if π ≤ π∗, ∗ α ∗ σ2(1−γβ) and P (π) = P if π > π , for j = L, H, some δ ∈ 0, 1−γβ , and π = δ + δ[1−γβ(2−β)] . This rule can be implemented with a maximally-enforced target threshold π∗.
The optimal target-based rule provides incentives to the central bank with a maximally-enforced target threshold π∗. Since a more expansionary monetary pol- icy µ results in higher inflation π in expectation, a central bank of type j responds to this threshold by choosing a policy sj +δ which is below its flexible policy µf (sj, Eµ) = sj + α + γβEµ. Note that punishment occurs along the equilibrium path whenever π > π∗, so as to appropriately incentivize the central bank. Since punishment is socially costly, the optimal target-based rule limits its fre- quency by keeping the central bank’s policy above the first-best level. That is, while a rule that induces the socially optimal policy with δ = 0 is available, Proposition 2 shows that this rule is strictly dominated by one that allows distortions with δ > 0. The proposition also shows that the induced expected inflation is below the threshold, i.e., E (π) = δ < π∗. A rule that yields E(π) = δ = π∗ would be suboptimal, as it would entail punishing the central bank half of the time (the frequency with which π would exceed π∗). In the optimal rule, realized inflation π exceeds π∗ less than half of the time so that punishment occurs less often. To prove Proposition 2, we follow a first-order approach and solve a relaxed version of the program in (16) that replaces the incentive compatibility constraint (13) with the first-order condition (18) of the central bank’s problem. Specifically, we consider a doubly-relaxed problem that takes (18) as a weak inequality constraint (cf. Rogerson, 1985) in order to establish the sign of the Lagrange multiplier on (18) and characterize the solution. We prove that the solution to the relaxed problem takes the threshold form described above, and we show that Assumption 1 and Assumption 2 are sufficient to guarantee the validity of this first-order approach. High-powered incentives of the form described in Proposition 2 arise in moral hazard settings where, as in our model, rewards and punishments are bounded and enter welfare linearly; see for example Innes(1990) and Levin(2003). These incentives arise here because punishment cannot directly depend on the central bank’s policy under a target-based rule. In Section 5, we show that Proposition 2 remains valid when the central bank’s signal is not binary but drawn from a continuum: since the central bank’s first-order condition implies that social welfare is independent of the
17 central bank’s type, social welfare is also independent of the number of central bank types.
3.3 Optimal Class of Rule
Our main result uses the characterizations in Proposition 1 and Proposition 2 to compare the performance of instrument-based and target-based rules. We find that which class of rule is optimal for society depends on the precision of the central bank’s private information:
Proposition 3. Take instrument-based and target-based rules and consider changing σ while keeping V ar(θ) unchanged. There exists σ∗ > 0 such that a target-based rule is strictly optimal if σ < σ∗ and an instrument-based rule is strictly optimal if σ > σ∗. The cutoff σ∗ is decreasing in the central bank’s bias α and increasing in the worst punishment P .
To see the logic, consider how social welfare under each class of rule changes as we vary the precision of the central bank’s information σ−1, while keeping the shock variance V ar(θ) unchanged. Since the optimal instrument-based rule gives no flexi- bility to the central bank to use its private information, social welfare under this rule 2 is invariant to σ. Specifically, by Proposition 1 and V ar(θ) = E(θ ) (since E(θ) = 0), social welfare under the optimal instrument-based rule is given by
V ar(θ) − , 2 independent of σ.29 In contrast, using Proposition 2, we can verify that social welfare under the optimal target-based rule is decreasing in σ, that is increasing in the preci- sion of the central bank’s information. Intuitively, a better informed central bank can more closely tailor its policy to the shock, and is less likely to trigger punishment by overshooting the inflation threshold specified by society. As a result, a higher preci- sion reduces the volatility of inflation and the social cost of providing high-powered incentives under a target-based rule. These comparative statics imply that to prove the first part of Proposition 3, it suffices to show that a target-based rule is optimal for high enough precision of the central bank’s information whereas an instrument-based rule is optimal otherwise.
29This independence will not extend to the setting of Section 5 with a continuum of signals, yet our main results will continue to apply.
18 Consider the extreme in which the central bank is perfectly informed, that is, σ → 0 and ∆ → pV ar(θ). In this case, the optimal target-based rule sets a threshold π∗ = 0, providing steep incentives and inducing the first-best policy. Note that this rule involves no punishments along the equilibrium path, as a perfectly informed central bank of type j = L, H chooses µj = µfb(sj) = sj to avoid punishment. Consequently, in this limit case, the optimal target-based rule yields social welfare
V ar(θ) 0 > − , 2 and thus it dominates the optimal instrument-based rule. Consider next the extreme in which the central bank is uninformed, that is, σ → pV ar(θ) and ∆ → 0. In this case, the optimal instrument-based rule guarantees the socially optimal policy given no information by tying the hands of the central bank. Instead, a target-based rule gives the central bank unnecessary discretion and requires punishments to provide incentives. The optimal target-based rule in this limit case sets a threshold π∗ > 0, inducing a central bank of type j = L, H to choose µj = sj +δ for δ > 0 and yielding social welfare
V ar(θ) δ2 V ar(θ) − − − Φ δ − π∗|0, σ2 P < − . 2 2 2
Thus, this rule is dominated by the optimal instrument-based rule. The second part of Proposition 3 shows that the benefit of using a target-based rule over an instrument-based rule is decreasing in the central bank’s inflation bias and increasing in the severity of punishment. The less biased is the central bank, the less costly is incentive provision under a target-based rule, as relatively infrequent punishments become sufficient to deter excessively expansionary policy. Similarly, the harsher is the punishment experienced by the central bank for missing the target threshold, the less often punishment needs to be used on the equilibrium path to provide incentives under a target-based rule. In contrast, the optimal instrument- based rule is independent of the central bank’s bias and the severity of punishment. As such, target-based rules dominate instrument-based rules for a larger range of parameters when the central bank’s bias is relatively small or punishment is relatively severe.30 30One may also wonder how the comparison of the two rule classes changes if the unconditional variance of the shock increases. The answer is ambiguous in that it depends on the extent to which the increase in V ar(θ) is due to an increase in σ or ∆: if only σ increases, instrument-based rules
19 In addition to the results in Proposition 3, the comparison of instrument-based and target-based rules reveals differences with regards to volatility, specifically to how the two rule classes trade off different kinds of volatility. Relative to the optimal target-based rule, the optimal instrument-based rule induces a lower variance of the central bank’s policy, while yielding relatively more volatile output and inflation. At the same time, the optimal instrument-based rule yields lower average inflation than the optimal target-based rule.
4 Optimal Unconstrained Rule
A hybrid rule combines features of instrument-based and target-based rules, with a punishment P (µ, µ − θ) that depends freely on the central bank’s policy choice µ and inflation µ − θ. For j = L, H, denote by P j(θ) the punishment assigned to central bank type j as a function of the shock θ. Analogous to the program in (12), an optimal hybrid rule solves:
Z ∞ X 1 j j j 2 max U(µ , θ, Eµ) − P (θ) φ(θ|s , σ )dθ (19) µL,µH ,{P L(θ),P H (θ)} 2 θ∈R j=L,H −∞
subject to, for j = L, H,
Z ∞ j j j j 2 αµ + U(µ , θ, Eµ) − P (θ) φ(θ|s , σ )dθ −∞ Z ∞ −j −j −j j 2 ≥ αµ + U(µ , θ, Eµ) − P (θ) φ(θ|s , σ )dθ, (20) −∞
Z ∞ j j j j 2 αµ + U(µ , θ, Eµ) − P (θ) φ(θ|s , σ )dθ −∞ Z ∞ f j f j j 2 ≥ αµ (s , Eµ) + U(µ (s , Eµ), θ, Eµ) − P φ(θ|s , σ )dθ, (21) −∞
j P (θ) ∈ [0, P ] for all θ ∈ R, and (22)
1 1 µ = µL + µH . (23) E 2 2 become more beneficial over target-based rules (as implied by Proposition 3), whereas if only ∆ increases, the opposite is true.
20 The solution to this program yields the highest social welfare that can be achieved subject to the central bank being inflation-biased and privately informed. Note that constraints (20)-(21) are analogous to (8)-(9) in the program that solves for the optimal instrument-based rule, but they now allow the punishment to depend on the shock θ in addition to the central bank’s policy µj. Note also that analogous to our analysis of the optimal target-based rule, the punishment P j(θ) can be equivalently written as a function of inflation, P j(π). We use this formulation in what follows to ease the interpretation. ∗ ∗∗ ∗ Define a maximally-enforced hybrid threshold {µ , µ , {π (µ)}µ∈R} as a rule that specifies no punishment if inflation is weakly below a threshold π∗(µ) and the maximal punishment P if inflation exceeds π∗(µ), where this threshold depends on the central bank’s policy µ and satisfies
∞ if µ ≤ µ∗ π∗(µ) = h (µ) if µ ∈ (µ∗, µ∗∗] (24) −∞ if µ > µ∗∗
∗ ∗∗ for cutoffs µ < µ and some continuous function h (µ) ∈ (−∞, ∞) with limµ↓µ∗ h (µ) = ∞. The cutoff µ∗ is a soft instrument threshold, where any policy µ ≤ µ∗ is not pun- ished independently of inflation. The cutoff µ∗∗ > µ∗ is a hard instrument threshold, where any policy µ > µ∗∗ is maximally punished independently of inflation. Inter- mediate policies µ ∈ (µ∗, µ∗∗] are not punished if inflation satisfies π ≤ π∗ (µ) and maximally punished if inflation satisfies π > π∗ (µ). Therefore, an interior target threshold only applies to intermediate policy choices. We find:
Proposition 4. The optimal hybrid rule specifies µL < µH , P L(π) = 0 for all π, P H (π) = 0 if π ≤ π∗(µH ), and P H (π) = P if π > π∗(µH ), for some π∗(µH ) ∈ (−∞, ∞). This rule can be implemented with a maximally-enforced hybrid threshold ∗ ∗∗ ∗ ∗ L ∗∗ H {µ , µ , {π (µ)}µ∈R}, where µ = µ and µ = µ .
The optimal hybrid rule prescribes a tight monetary policy and no punishment for the low central bank type, while specifying an expansionary monetary policy and a target threshold for the high central bank type. To prove this result, we solve a relaxed version of (19)-(23) which ignores the private information constraint (20) for the high type and the enforcement constraints (21) for both types. We establish that
21 the solution to this relaxed problem takes the form described in Proposition 4 and satisfies these constraints. As implied by Proposition 4, the optimal hybrid rule strictly improves upon the rules studied in Section 3. Intuitively, this rule dominates instrument-based rules by giving the central bank more flexibility to respond to its private information while preserving incentives. The reason is that, under a hybrid rule, society can allow the central bank to choose policies µ > µ∗ and still deter excessively expansionary policies by using a target-based criterion. Analogously, the optimal hybrid rule dominates target-based rules by more efficiently limiting the central bank’s discretion to choose policies that are excessively loose. The reason is that, under a hybrid rule, society can avoid punishments under policies µ ≤ µ∗ and directly punish the central bank for policies µ > µ∗∗, thus reducing the frequency of punishment on path. In principle, combining instruments and targets could yield rules with complicated forms. Proposition 4 however shows that the optimal hybrid rule admits an intuitive implementation. This rule essentially consists of an instrument threshold µ∗ which is relaxed to µ∗∗ whenever the target threshold is satisfied. Interestingly, as discussed in the Introduction, monetary policy rules of this form were advocated in the US in the aftermath of the Global Financial Crisis.31
5 Continuum of Types
Our baseline model facilitates a transparent analysis by assuming that the central bank’s signal of the demand shock is binary. In this section, we show that our main insights remain valid under a continuum of signals. Suppose that instead of observing a binary signal s ∈ {−∆, ∆} of the demand shock θ ∈ R, the central bank observes a signal as rich as the shock: s ∈ R with 2 2 s ∼ N (0, ∆ ). The shock’s conditional distribution obeys θ|s ∼ N (s, σ ), and thus the unconditional distribution has E(θ) = 0 and V ar(θ) = σ2 + ∆2 as in our baseline model. We define the optimal instrument-based and target-based rules analogously as in Section 3, with formal representations provided in the Online Appendix. In what follows, we take γ ∈ {0, 1}. The case of γ = 0 corresponds exactly to the broadly studied delegation setting with quadratic preferences and a constant bias, as in Melumad and Shibano(1991) and Alonso and Matouschek(2008) among others. The
31The optimal hybrid rule also resembles the monetary policy adopted by the Italian government in the 1990s; see fn. 11.
22 case of γ = 1 instead brings the model close to a New Keynesian setting with negligible costs of inflation, as suggested by the work of Nakamura et al.(2018). In both cases, we can show that the expectations constraint—namely the analog of constraint (11) under an instrument-based rule and correspondingly constraint (15) under a target- based rule—is no longer binding. This simplifies the analysis and allows us to derive the following characterization.32
Proposition 5. Consider an environment with a continuum of central bank types and γ ∈ {0, 1}. Then:
1. An optimal instrument-based rule specifies s∗, s∗∗ ∈ (−∞, ∞) such that
µf (s, µ), 0 if s < s∗, E f ∗ ∗ ∗∗ {µ(s),P (s)} = µ (s , Eµ), 0 if s ∈ [s , s ], f ∗∗ µ (s, Eµ), P if s > s ,
where
Z ∞ f ∗ f ∗ ∗∗ 2 αµ (s , Eµ) + U(µ (s , Eµ), θ, Eµ) φ(θ|s , σ )dθ −∞ Z ∞ f ∗∗ f ∗∗ ∗∗ 2 = αµ (s , Eµ) + U(µ (s , Eµ), θ, Eµ) − P φ(θ|s , σ )dθ. −∞
This rule can be implemented with a maximally-enforced instrument threshold µ∗ = µf (s∗, Eµ).
2. An optimal target-based rule specifies µ(s) = s + δ, P (π) = 0 if π ≤ π∗, and ∗ α ∗ σ2(1−γβ) P (π) = P if π > π , for some δ ∈ 0, 1−γβ and π = δ + δ[1−γβ(2−β)] . This rule can be implemented with a maximally-enforced target threshold π∗ which coincides with that in Proposition 2.
The optimal instrument-based and target-based rules under a continuum of cen- tral bank types take the same implementations as under binary types. There are differences, however, with regards to the central bank’s behavior under the optimal instrument-based rule. In the case of binary types, Proposition 1 shows that both types of the central bank are bunched at the threshold µ∗, with no discretion and with
32We assume that an optimal instrument-based rule is piecewise continuously differentiable. Also, if the program solving for this rule admits multiple solutions that differ only on a countable set of types, we select the solution that maximizes social welfare for those types.
23 no punishments on path. These features change with a continuum of types. Propo- sition 5 shows that in this case, central bank types s ∈ [s∗, s∗∗] are bunched at the threshold µ∗ and assigned no punishment. But central bank types s < s∗ are given discretion: they satisfy the instrument threshold strictly by choosing their flexible policy µf (s, Eµ) while enjoying no punishment. Moreover, central bank types s > s∗∗ are punished on path: these types break the instrument threshold by choosing their flexible policy µf (s, Eµ) > µ∗ and are assigned the maximal punishment P . The cut- off s∗∗ corresponds to the type that is indifferent between abiding to the instrument threshold and receiving no punishment versus breaking the threshold and receiving maximal punishment, as shown by the indifference condition in Proposition 5. Compared to the analysis in Subsection 3.1, characterizing the optimal instrument- based rule under a continuum of types requires more involved arguments. These arguments are related to those developed in Halac and Yared(2019), and we present them in the Online Appendix in three main steps.33 First, we show that the linearity of social welfare and central bank welfare in the punishment, together with the richness of the information structure, imply that only zero and maximal punishment P are prescribed in an optimal instrument-based rule; that is, optimal punishments are bang- bang. Second, appealing to the log concavity of the normal density function, we establish that optimal punishments are also monotonic, with central bank types below some cutoff s∗∗ receiving zero punishment and central bank types above s∗∗ receiving maximal punishment P .34 As an implication, types s > s∗∗ must choose their flexible policy µf (s, Eµ). Finally, we show that for s ≤ s∗∗, the optimal rule prescribes policy that is continuous in the central bank’s type and takes the form of a threshold. The characterization of the optimal target-based rule under a continuum of types follows the same steps as in Subsection 3.2. In fact, Proposition 5 shows that the optimal target threshold π∗ is quantitatively identical to that in the case of binary types. As mentioned in Subsection 3.2, the reason is that the first-order condition of the central bank’s problem in (18) implies a choice δ = µ − s by the central bank which is independent of its type. This means that the optimal target-based rule is independent of the distribution of the central bank’s signal s, and therefore the
33The arguments in Halac and Yared(2019) cannot be applied directly to the current setting as the agent’s bias in that paper takes a multiplicative form. We provide a complete proof in the Online Appendix for the additive bias specification of this paper. 34We note that our proof in the Online Appendix, and thus the characterization of the optimal instrument-based rule in Proposition 5, apply more broadly to any distribution of types with a log concave density. This includes the exponential, gamma, log-normal, Pareto, and uniform distributions among others; see Bagnoli and Bergstrom(2005).
24 characterization in Proposition 2 holds independently of the number of types. The next proposition uses the characterizations in Proposition 5 to compare the performance of instrument-based and target-based rules. We show that our main result in Proposition 3 continues to apply under a continuum of types: target-based rules dominate instrument-based rules if and only if the central bank’s private information is sufficiently precise.
Proposition 6. Consider an environment with a continuum of central bank types and γ ∈ {0, 1}. Take instrument-based and target-based rules and consider changing σ while keeping V ar(θ) unchanged. There exists σ∗ > 0 such that a target-based rule is strictly optimal if σ < σ∗ and an instrument-based rule is strictly optimal if σ > σ∗.
The result follows from comparing how an increase in the precision of the central bank’s signal affects social welfare under the optimal instrument-based rule versus the optimal target-based rule. The argument is more involved than in the case of binary types, as now a more precise signal strictly improves social welfare under both classes of rules. Specifically, in the case of the optimal target-based rule, the effect takes the same form as in Subsection 3.3: if the central bank’s information is more precise, then the central bank can better tailor its policy to the shock, which lowers inflation volatility and thus also the need to utilize costly punishments on path. In the case of the optimal instrument-based rule, the effect is richer than in Subsection 3.3: a higher signal precision now allows a better tailoring of the policy to the shock for central bank types s < s∗ and s > s∗∗ (for cutoffs s∗ and s∗∗ defined in Proposition 5), therefore lowering inflation volatility under this rule too. We prove that the social welfare effect of making the central bank’s signal more precise is smaller under the optimal instrument-based rule than under the optimal target-based rule. Intuitively, while inflation volatility declines with precision for types s < s∗ and s > s∗∗ under the instrument-based rule, there are no-target based pun- ishments whose frequency is reduced as a result; moreover, a higher signal precision has no effect on types s ∈ [s∗, s∗∗] who have no discretion under this rule. In contrast, under the target-based rule, inflation volatility declines with precision for all types, and this has the additional effect of reducing target-based punishments on path. Con- sequently, we are able to show that the relative benefit of using a target-based rule increases with the precision of the central bank’s signal. This comparison in turn al- lows us to obtain the result in Proposition 6 and hence to establish the robustness of our findings to a continuum of central bank types.
25 6 Extensions
We discuss three extensions of our baseline model. In the first extension, we relax the assumption that punishments are common to the central bank and society by considering asymmetric punishment costs. In the second extension, we study what happens when the shocks affecting the economy are not i.i.d. but persistent over time. Finally, in the third extension, we consider other policy instruments that may be available to the central bank in addition to the interest rate. In all of these, we show that our main findings continue to hold.
6.1 Asymmetric Punishments
Society may be able to specify punishments that harm the central bank more severely than society itself. Consider an extension of our baseline model in which varying the punishment has a larger effect on the central bank’s welfare compared to social welfare. Specifically, let us modify the central bank’s welfare in (6) so that it is now given by
Z ∞ j j j j j 2 α µ − θ + U(µ , θ, Eµ) − cP (µ , µ − θ) φ(θ|s , σ )dθ (25) −∞ for j = L, H and some c ∈ (1, α/2∆).35 Note that our baseline model corresponds to setting c = 1. Under c > 1, any punishment P (µj, µj − θ) implies a larger cost on the central bank than on society. This asymmetric-punishments setting yields optimal instrument-based and target- based rules which take the same implementations as under symmetric punishments. The optimal instrument-based rule is in fact quantitatively identical to that in Sub- section 3.1, with µL = µH = 0 and P L = P H = 0. Intuitively, given c < α/2∆, it is still the case that providing incentives to separate the two central bank types is too costly for society, so the optimal instrument-based rule continues to bunch both types. The optimal target-based rule takes the same form as in Subsection 3.2, with high-powered incentives being optimal as the central bank’s welfare remains linear in the punishment under (25). Nonetheless, this rule is not quantitatively identical to that under symmetric punishments. The reason is intuitive: since imposing any given punishment on the central bank, and therefore providing incentives, is now less costly for society, the optimal target-based rule induces a policy closer to first best (i.e., with
35Our modeling of asymmetric punishments follows Amador and Bagwell(2013) in their study of delegation under imperfect transfers.
26 a lower value of δ) in the case of asymmetric punishments. Our main result in Proposition 3 also extends to this asymmetric-punishments setting: target-based rules dominate instrument-based rules if and only if the central bank’s private information is sufficiently precise. The argument is the same as in the case of symmetric punishments. In particular, note that since the optimal instrument- based rule provides no discretion to the central bank, social welfare under this rule is invariant to the precision of the central bank’s information, as in Subsection 3.3. Moreover, since the optimal target-based rule takes the same form as under symmetric punishments, social welfare under this rule increases with the precision of the central bank’s information, also as in Subsection 3.3. Based on these comparative statics, it follows that our main insights are robust to asymmetric punishments.
6.2 Persistent Shocks
We have assumed that the demand shock θt is i.i.d. over time. We next show that our results remain valid if the shock is instead persistent. In fact, we find that by suitably reformulating society’s problem, our analysis can be applied without change. We model persistent shocks by letting the realization of the shock at time t−1 shift the mean of the shock at time t. That is, suppose E[θt|θt−1] = f(θt−1) for a weakly increasing function f and t = 1, 2,..., and let the signal st also be shifted by θt−1 so that st ∈ {f(θt−1) − ∆, f(θt−1) + ∆}. As in our baseline model, we assume that each 2 signal is realized with equal probability and θt|st ∼ N (st, σ ). We focus our attention on Markov perfect equilibria in which the central bank’s strategy at date t depends only on st and the punishment function Pt(·) specified by society. Furthermore, we consider equilibria within this class in which the distributions of inflation and the output gap in every period are constant (while the distribution of future interest rates may vary over time). These criteria are desirable in that they are satisfied in the first-best allocation, which features zero expected inflation, a zero expected output gap, and an interest rate that co-moves with the signal st. To show that an equilibrium as just defined exists, let Eπ and Ex denote the time- invariant means of inflation and the output gap respectively. We can define µet in this environment to be analogous to µt in our baseline model, except for the mean of θt which is now given by f(θt−1) instead of zero:
µet ≡ f(θt−1) + (β + κζ)Eπ + κEx − κζit.
27 Additionally, define eθt ≡ θt −f(θt−1) and set ≡ st −f(θt−1). Note that the distributions of eθt and set are i.i.d. and the same as in our baseline model. It follows by substitution into (1) and (2) that πt = µet −eθt, Eπt = Eµet, and xt = (µet −eθt − βEµet)/κ. Therefore, πt and xt are guaranteed to be stationary if µet is stationary, in which case we can ignore time subscripts. The punishment function specified by society is P (µ,e µe − eθ), and per-period social welfare can be represented as
X 1 Z ∞ [U(µj, eθ, µ) − P (µj, µj − eθ)]φ(eθ|sj, σ2)dθ 2 e Ee e e e j=L,H −∞ 2 j j 2 j (µ − eθ − βEµ) (µ − eθ) where U(µ , eθ, µ) = −γ e e − (1 − γ) e . e Ee 2 2
j j j Finally, given a signal s and taking as given the punishment function P (µe , µe − eθ) j and expectations Eµe, the central bank chooses a policy µ to maximize Z ∞ j j j j j 2 [α(µe − eθ) + U(µe , eθ, Eµe) − P (µe , µe − eθ)]φ(eθ|se , σ )dθ −∞ for j = L, H. With the variables and objective functions reformulated in this way, it is clear that we can apply the analysis from our baseline model without further change. Consequently, all of our results remain valid in this environment with persistent de- mand shocks. It is worth noting that, in this environment, the implementation of the optimal instrument-based rule as a maximally-enforced lower bound on the nominal interest rate would entail a bound that varies with the state θt−1.
6.3 Other Instruments
In our baseline model, the central bank has access to only one policy instrument, namely the interest rate. We next discuss two extensions of our environment in which other policy instruments are available. First, suppose that instead of studying a closed economy, we consider an open- economy New Keynesian setting (e.g. Woodford, 2003; Gal´ı, 2015). Such a setting is characterized by the Phillips curve in (1), the Euler equation in (2), and an uncovered interest rate parity condition given by
∗ it = it + Et(∆et+1), (26)
28 ∗ where it is the domestic interest rate, it is the interest rate abroad, and ∆et+1 is the ∗ exchange rate devaluation. Assume that it is known at date t when the central bank makes its policy choice. Then by (26), a choice of exchange rate devaluation ∆et+1 by the central bank implies a choice of domestic interest rate it. Since welfare in this economy depends on inflation πt and the output gap xt exactly as in our baseline model, it follows that our analysis applies: the optimal instrument-based rule can be implemented as a maximally-enforced lower bound on the domestic interest rate or, equivalently, a maximally-enforced upper bound on exchange rate devaluation. Second, suppose we consider a closed-economy setting with money growth. Such a setting is characterized by the Phillips curve in (1), the Euler equation in (2), and a money demand equation that can be represented dynamically in terms of money growth ∆mt (e.g., Woodford, 2003; Gal´ı, 2015):
∆mt = πt + xt − ηit − xt−1 + ηit−1, (27)
for η > 0. Given it−1 and given equations (1) and (2) for πt and xt respectively, a choice of money growth ∆mt by the central bank implies a monetary stance that maps into a level of inflation and output gap. Thus, as above, this environment is analogous to our baseline model and our analysis applies: the optimal instrument-based rule puts a cap on how expansionary monetary policy can be, so it can be implemented as a maximally-enforced upper bound on money growth.
7 Concluding Remarks
In this paper, we have studied the optimal design of monetary policy rules. We considered how to provide incentives to a central bank which has non-contractible information about shocks to the economy and cannot commit to policy. Using a mechanism design approach, we characterized optimal instrument-based and target- based rules, examined the conditions under which each rule class is optimal, and solved for the optimal unconstrained rule that combines instruments and targets. Our results imply that inflation targeting should be adopted if the central bank’s information is sufficiently precise and its commitment problem is not too severe; otherwise an interest rate rule would perform better. This same conclusion applies in the context of other policy targets, such as output gap or price level targets (which are equivalent to an inflation target in our model), as well as in the context of other policy instruments,
29 such as an exchange rate or a monetary growth rate (as discussed in Subsection 6.3). We presented a stylized model with binary signals and symmetric punishments, and we showed that our main results extend to settings with a continuum of signals or asymmetric punishments. Our analysis could be further extended in various direc- tions. One extension could explore different specifications of asymmetric punishments by allowing for non-linearities. For example, while society may suffer a lower cost than the central bank from any given punishment, as in the setting discussed in Subsec- tion 6.1, society’s cost may also increase at a relatively faster rate with the severity of punishment, becoming closer to the central bank’s cost at higher levels. Note that different specifications could give rise to different penal codes. In particular, unlike the “bang-bang” punishments that we found to be optimal in our model, the solution could feature punishment that “fits the crime.” Another extension could consider a central bank whose inflation bias is unknown to society, possibly admitting not only different levels but also different signs. Under binary signals and assumptions analogous to Assumption 1 and Assumption 2, an instrument-based rule that bunches the central bank types regardless of their bias would be optimal, so our characterization in Proposition 1 would remain valid. A characterization of the optimal target-based rule, on the other hand, would have to deal with the problem that assigned policies may now depend on the central bank’s unknown bias. While our focus has been on monetary policy—an area where both instruments and targets have been widely used in practice and extensively discussed in the literature— our analysis also holds lessons for other applications.36 For example, in the context of fiscal policy, rules may place constraints on instruments like spending or on targets like deficits (see National Conference of State Legislatures, 1999). Bohn and Inman(1996) document that a number of US states use rules based on instruments in the form of beginning-of-the-year fiscal requirements, whereas other US states use a target-based criterion in the form of end-of-the-year fiscal requirements. Our analysis suggests that target-based rules will be relatively more favorable if the government has sufficiently precise non-contractible information about future revenues and its deficit bias is not too large. Moreover, a hybrid rule specifying a spending cap that switches to a deficit
36The questions we study are relevant for the design of environmental policies. Environmental regulation may focus on technology mandates—requirements on firms’ production processes, such as the choice of equipment—or on performance standards—requirements on output, such as maximum emission rates. See, for example, US Congress, Office of Technology Assessment(1995) and Goulder and Parry(2008).
30 target when violated would dominate both instrument-based and target-based fiscal policy rules. Notably, such a hybrid rule resembles the rule underlying the Swiss debt brake (Beljean and Geier, 2013).
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A Proofs for Section 3 and Section 4
This Appendix provides proofs for the results in Section 3 and Section 4. See the Online Appendix for proofs of the results in Section 5.
A.1 Proof of Proposition 1
We proceed in three steps.
Step 1. We solve a relaxed version of the program in (12) which ignores (8) for j = H and (9) for j = L, H. Step 2 verifies that the solution to this relaxed problem satisfies these constraints.
Step 1a. We show that the solution to the relaxed problem satisfies (8) for j = L as an equality. If this were not the case, then the solution would set µj = sj and P j = 0 for j = L, H. However, (8) for j = L would then become
Z ∞ Z ∞ αsL + U(sL, θ, 0) φ(θ|sL, σ2)dθ ≥ αsH + U(sH , θ, 0) φ(θ|sL, σ2)dθ, −∞ −∞ which after some algebra yields